Complex roots, conjugated complex numbers

Question

Knowing that $$\cos\frac{\pi}{8}=\frac {1}{2}\sqrt{2+\sqrt{2}},$$ find all roots of these equations:

$2 \overline z=z^7$,

$32 \overline z=z^7$,

$128 \overline z+z^7=0$.

Only those which have solutions different from $z=0$.

Answer 1

For the first one we have that

$$2\overline z=z^7 \implies 2\overline zz=z^8 \implies z^8=2|z|^2\implies |z|^6=2 \quad z=\sqrt[6] 2$$

then we need to solve

$$z^8=2\sqrt[3] 2$$

and similarly for the others.

The solution seems not related to $\cos \frac{\pi}8$.